A272479 a(n) is the smallest k different from n such that (n, k) is a Harshad amicable pair (see the comments).

10, 10, 12, 20, 10, 12, 70, 40, 18, 1, 10, 3, 76, 10, 12, 35, 296, 9, 10, 2, 3, 20, 10, 6, 14, 184, 9, 10, 20999, 3, 100, 10, 12, 98, 16, 9, 10, 11, 12, 4, 10, 6, 99799, 40, 9, 10, 2099999, 12, 52, 5, 12, 49, 1000, 9, 10, 11, 12, 1001, 7998998, 6, 7999999, 200, 9, 10, 319, 12, 68989999, 98, 30, 7, 1000, 9, 10, 11, 12, 13, 56, 15, 10000, 8

Offset

1,1

Comments

Let m and k be distinct integers and dsum(n) be the sum of digits of n. We call m and k Harshad amicable if dsum(m) divides k and dsum(k) divides m.

For any n with no Harshad amicable partner, a(n)=0.

Conjecture: the sequence contains no zeros.

Large terms of a(n) correspond to prime indices and prime indices whose sum of digits is prime correspond to particularly large terms.

Links

Table of n, a(n) for n=1..80.

Example

For n=12, a(12)=3 as the smallest number such that its sum of digits (3) divides n and the sum of digits of n (3) divides a(n).
For n=13, a(13)=76 as the smallest number such that its sum of digits (13) divides n and the sum of digits of n (4) divides a(n).

Mathematica

lst={}; Do[k=1; While[ k!=n&& !(Divisible[n, Total@IntegerDigits@k]&& Divisible[k, Total@IntegerDigits@n]), k++]; If[k==n, k=n+1; While[!(Divisible[n, Total@IntegerDigits@k]&& Divisible[k, Total@IntegerDigits@n]), k++]]; AppendTo[lst, k], {n, 1, 80}]; lst

Prog

(PARI) for(n=1, 80, k=1; while(k!=n && !(n%sumdigits(k)==0 && k%sumdigits(n)==0), k++); if(k==n, k=n+1; while(!(n%sumdigits(k)==0 && k%sumdigits(n)==0), k++)); print1(k ", "))

Crossrefs

Cf. A005349 (Harshad numbers), A007953 (digital sum).

Sequence in context: A354113 A354114 A175220 * A167427 A227437 A352314

Adjacent sequences: A272476 A272477 A272478 * A272480 A272481 A272482

Keyword

nonn,base

Author

Waldemar Puszkarz, May 01 2016

Status

approved